Kuranishi Structures and Virtual Fundamental Chains

1. Front Matter

   Pages i-xv

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2. Introduction

   • Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono

   Pages 1-39

3. Notations and Conventions

   • Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono

   Pages 41-45

4. Abstract Theory of Kuranishi Structures, Fiber Products and Perturbations

   1. Front Matter

      Pages 47-47

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   2. Kuranishi Structures and Good Coordinate Systems

      • Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono

      Pages 49-65

   3. Fiber Product of Kuranishi Structures

      • Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono

      Pages 67-77

   4. Thickening of a Kuranishi Structure

      • Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono

      Pages 79-94

   5. Multivalued Perturbations

      • Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono

      Pages 95-111

   6. CF-Perturbations and Integration Along the Fiber (Pushout)

      • Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono

      Pages 113-146

   7. Stokes’ Formula

      • Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono

      Pages 147-157

   8. From Good Coordinate Systems to Kuranishi Structures and Back with CF-Perturbations

      • Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono

      Pages 159-176

   9. Composition Formula of Smooth Correspondences

      • Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono

      Pages 177-192

   10. Construction of Good Coordinate Systems

       • Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono

       Pages 193-219

   11. Construction of CF-Perturbations

       • Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono

       Pages 221-237

   12. Construction of Multivalued Perturbations

       • Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono

       Pages 239-251

   13. Zero-and One-Dimensional Cases via Multivalued Perturbation

       • Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono

       Pages 253-273

5. System of K-Spaces and Smooth Correspondences

   1. Front Matter

      Pages 275-275

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   2. Introduction to Part II

      • Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono

      Pages 277-311

   3. Linear K-Systems: Floer Cohomology I – Statement

      • Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono

      Pages 313-343

   4. Extension of a Kuranishi Structure and Its Perturbation from Boundary to Its Neighborhood

      • Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono

      Pages 345-398

The package of Gromov’s pseudo-holomorphic curves is a major tool in global symplectic geometry and its applications, including mirror symmetry and Hamiltonian dynamics. The Kuranishi structure was introduced by two of the authors of the present volume in the mid-1990s to apply this machinery on general symplectic manifolds without assuming any specific restrictions. It was further amplified by this book’s authors in their monograph Lagrangian Intersection Floer Theory and in many other publications of theirs and others. Answering popular demand, the authors now present the current book, in which they provide a detailed, self-contained explanation of the theory of Kuranishi structures.


Part I discusses the theory on a single space equipped with Kuranishi structure, called a K-space, and its relevant basic package. First, the definition of a K-space and maps to the standard manifold are provided. Definitions are given for fiber products, differentialforms, partitions of unity, and the notion of CF-perturbations on the K-space. Then, using CF-perturbations, the authors define the integration on K-space and the push-forward of differential forms, and generalize Stokes' formula and Fubini's theorem in this framework. Also, “virtual fundamental class” is defined, and its cobordism invariance is proved.


Part II discusses the (compatible) system of K-spaces and the process of going from “geometry” to “homological algebra”. Thorough explanations of the extension of given perturbations on the boundary to the interior are presented. Also explained is the process of taking the “homotopy limit” needed to handle a system of infinitely many moduli spaces. Having in mind the future application of these chain level constructions beyond those already known, an axiomatic approach is taken by listing the properties of the system of the relevant moduli spaces and then a self-contained account of the construction of the associated algebraic structures is given. This axiomatic approach makes the exposition contained here independent of previously published construction of relevant structures. 

• Virtual fundamental chain
• Kuranishi structure
• Floer homology
• A infinity structure
• Pseudo holomorphic curve

“Although the authors have dealt with this subject in some previous monographs and in numerous articles, it seemed necessary to have a book that could collect all the necessary and well-organized material. This is the reason why the two above-mentioned authors, together with Yong-Geun Oh and Hiroshi Ono, have decided to write this book that we review now. … this book (an authentic tour de force) it is an indispensable reference for every researcher in symplectic geometry and topology.” (Manuel de León, zbMATH 1482.53002, 2022)

• Simons Center for Geometry and Physics, State University of New York, Stony Brook, USA

  Kenji Fukaya

• IBS Center for Geometry and Physics Pohang, Gyung-Buk, Korea (Republic of)

  Yong-Geun Oh

• Graduate School of Mathematics Nagoya University, Nagoya, Japan

  Hiroshi Ohta

• Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan

  Kaoru Ono

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